My theory - in brief form

I've posted this theory elsewhere, so unless someone wants a real thorough exposition on it I won't go into all the details. If you've already read this elsewhere, I don't plan on addressing this here very long.
I've found a way to express prime numbers that seems (to me) to have the property of synergy (synergy: the whole is more than the sum of the parts, etc.). I will give the theory, briefly, but my question (I didn't ask this question where I posted my theory before) is can anyone think of a way to apply the theory to other synergistic systems? For example, we put bread, sauce, and cheese together and call it pizza. Our tastebuds combine the experience of these flavors in a synergistic fashion to give us an experience that is new i.e. we might not have been able to predict our reaction to the combination based only on having eaten spoonfuls of sauce and bites of bread and cheese separately. Somehow, the combination has a property (enjoyment) that a spoonful of sauce just can't explain. This is synergy, the whole is more than the sum of the parts. My question is, can we somehow quantify the experiences of each part numerically, and then combine the numbers in a specific way that CAN predict the outcome of their combination? Btw, synergy plays a role not just in food, but psychology, economics, biology, physics, weather, artificial intelligence, any complex system with feedback loops.

Here's the math I've developed. Suppose we know ALL of the primes up to some value, say 2, 3, 5, 7, 11, 13 and that's all (I know we know a bunch more, but the theory's the same). Now square 13 to get 169 and set it aside for now. Take the rest of them and put them into two separate piles, say pile X is 2 and 7, and pile Y is 3, 5, and 11 (remember we set 13 to the side for now). Then we do this funky looking equation:
Q = abs( 2^a * 7^b +/- 3^c * 5^d * 11^e )
All this really means is take any exponents of all of the numbers in the two piles, then multiply the exponentiated numbers from the first pile together to get a number, say it is "A" and multiply the exponentiated numbers from the second pile to get "B", then do A plus or minus B, and if the result of the subtraction is negative take its absolute value. This will give two answers (one for plus and one for minus) for each choice of exponents a,b,c,d,e, so Q is actually a set of solutions. Oh, by the way, the exponents must all be integers greater than zero.
Now, my discovery is that if q is one of the solutions in set Q, and if q is less than 169 (remember the square of 13 that we set aside), then q will be prime. My conjecture (so far not disproven by anyone) is that ALL primes will be in the set Q. Of course, if you are looking for a prime bigger than 169 but less than, say, 529 which is the square of 23, then you will have to make two piles from all of the primes 2, 3, 5, 7, 11, 13, 17, 19 (and set 529 to the side for the "less than" test at the end), to make up your new set Q.
Now, I say the primes are synergistic because we are adding A and B, and the prime factors of A are some primes (2 and 7 in the example) and the prime factors of B are other primes (3, 5, 11 in the example) and all outputs less than 169 in Q are NEW primes BETWEEN 11 and 169. We have certain primes going in, and a set of NEW primes coming out - the whole is more than the sum of the parts - synergy. If you all think this is nonsense, I'll go down without a fight. But think for a moment: what if somehow we could express (or even arbitrarily assign) a certain quality (sauce-ness as percieved by a given person's tastebuds) as a prime, say "2", and then express another quality (bread-ness) as "3" and then the exponents would be the quantity of each that we're combining together in our pizza. Maybe certain outputs are favored - with 2 and 3 only, the primes output will be less than 25, so maybe people prefer smaller prime pizzas such as a "5 pizza" or a "7 pizza", or maybe people avoid pizzas that fit the prime scheme, only going for composite pizzas. Laugh at my example, perhaps, but we could also talk about artificial intelligence or the complex neural connections in the human brain this way. Isn't it true that neurons can each play different roles, depending on what "circuit" is activating them? Maybe each role has a different synergy. In physics and chemistry, some complex systems display entrainment or other emergent phenomenae. Who knows, really, how these things happen? This may not be the end-all theory, but could it be a start?
Aaron

Notice that in the example, not all primes less than 169 will be in Q. To get them all, it is necessary to divide the SAME SET of primes into piles A and B in different ways. We had 2 and 7 in pile A, the next step is to try 2 and 5 in A, or 2, 5, and 7 in A. After each new set of A and B is defined, we get a new Q. All of these Q's together will have all of the primes less than 169.

A quick example of how this all works:

try 3^1 * 5^1 * 7^1 +/- 2^x , where we are keeping the exponents on the 3, 5, and 7 constant for simplicity - this will give a subset of outputs in a Q where all members of Q that are less than 11^2=121 are prime.

Notice each output that passes the less-than 121 test is prime. The next step would be to increase the exponent on 3 or 5 or 7, then run the same process of exponentiating the 2. Keep doing this. Once you run out of viable outputs, make new A and B piles, then do it again.
Aaron

given any of the Q's (you don't mean solutions by the way, you're solving nothing), then none of its prime factors can be in the original list of primes, the set of Q's almost certainly contains all the other primes in the range (Euclid's algorthim, though I've not written the proof down), and only those (again the proof seems clear, though I've not written it down): if Q were composite it would be the product of at least two primes, both at least as big as the maximum in the list, but then it would be outside your valid range. Actually, thinking of it, yep, your theory is true for basic number theoretic considerations. I might ask some of my students to prove it; it's quite pretty.

Matt,
Thank you for the confirmation of the theory. Request: could you point me in the right direction with a little more detail on how to prove ALL primes in the range are in Q? In particular, we know that we've eliminated all composites of the form ax-bx=(a-b)x but we've also eliminated potential primes like 23-4=19, so we need to somehow show those primes are still in there (27-8=19) . It might be easy, but it's not obvious to me how to show that all primes are in Q.
Btw, if you use it for a class, you would be interested to know that the "+" is neither necessary, nor does it even give small enough solutions beyond a certain range (I think it was 169). But the subtraction is still enough, at least so far - I found representations of the proper form for all primes less than 1369 = 37^2.
Thanks again.
Aaron

Zero isn't allowed to be an exponent, though in one form "extra" primes can be added into the mix without losing anything from the proof except you won't be producing that particular prime in Q. But I'm more concerned with proving all primes in Q without these extra primes being put in A or B.

Yes, I believe 1 will appear in every Q, for every partitioning of the primes into sets A and B. I forgot to mention that. So it appears we are producing non-composites, instead of primes. It almost makes you think that 1 should be counted as a prime, but then alot of other theorems would have to begin by saying "this theorem deals with all primes except 1". Which would be okay with me. Notice my theory also works for 2^x +/- 1 and for 2^x * 3^y +/- 1, etc. Of course, like addition, the numbers on the left quickly get too big to produce small enough q in Q to guarantee the primality of q after subtracting 1.

One other thing I might mention, which you've probably noticed. If you ignore the "less-than" test you will still only get primes and composites that are relatively prime to AB, in other words this "bigger Q" will be a set "enriched" with primes. From that, another question presents itself: is every number relatively prime to AB in this bigger Q? The answer might be "no", but if it is "yes" that might help prove my first question. Thanks for the interest and inputs.
Aaron

For a long time 1 was considered a prime, but then as we made richer theories where there were more units (invertible elements) it became more useful to exclude 1 and say a prime was a number with exactly two factors

Thanks for the explanation of 1.
Any ideas on how to prove all primes in the range are in Q? I have worked for a long, long time (years) on it, but somehow the solution eludes me. I think I got "burned out" computing all primes < 1369, and now I have mathematical "writer's block" or something. It appears at first to be an easy question, but I think it may be more difficult than it looks. I'll look up Euclid's algorithm, it's been a while since I used it and I was never good at remembering it for some reason (I think I prefer reasoning out the solution like a computer programmer would do, over applying a pre-defined algorithm). But I think I looked at it before and it wasn't sufficient to prove this. If you can prove it, would you give me the first couple steps of a proof please? Appreciate the time you've taken with this.
Aaron

One last try, then I'll let it rest:
Does Anyone have Any ideas on how to prove all primes in a range are in Q? I'll take a suggestion of a half-baked possibility, anything may spark a creative solution. Probability and statistics (counting possible arrangements of differences
{19=20-1=21-2=22-3=23-4=24-5=25-6=26-7= (27-8) is a hit}
that lead to a target q and proving at least one such arrangement fits the given form by thinking of how many have factors of 3 in top or bottom, etc.) seems to me the most natural approach, but I have yet to get it quite right. Any suggestions at all?
Aaron

in the above post, what I meant by top and bottom is, for each pair A-B, examine how many pairs have a 3 in either A or B (two-thirds of them), and for larger primes we'd have to consider how many have 3 and 5, etc. Then we also have to consider how many DON'T have higher primes, and that's where I get stuck.